Rigidity theorems for complete lambda-hypersurfaces

摘要

In this article, we study hypersurfaces Sigma subset of Rn+1 with constant weighted mean curvature, also known as lambda-hypersurfaces. Recently, Wei-Peng proved a rigidity theorem for lambda-hypersurfaces that generalizes Le-Sesum's classification theorem for self-shrinkers. More specifically, they showed that a complete lambda-hypersurface with polynomial volume growth, bounded norm of the second fundamental form, and that satisfies vertical bar A vertical bar H-2(H - lambda) <= H-2/2 must either be a hyperplane or a generalized cylinder. We generalize this result by removing the bound condition on the norm of the second fundamental form. Moreover, we prove that under some conditions, if the reverse inequality holds, then the hypersurface must either be a hyperplane or a generalized cylinder. As an application of one of the results proved in this paper, we will obtain another version of the classification theorem obtained by the authors of this article, that is, we show that under some conditions, a complete lambda-hypersurface with H >= 0 must either be a hyperplane or a generalized cylinder.